High Pressure Mixing Rheology of Drilling Fluids - American Chemical

Article pubs.acs.org/IECR

High Pressure Mixing Rheology of Drilling Fluids J. Hermoso,† B. D. Jofore,‡ F. J. Martínez-Boza,*,† and C. Gallegos† †

Departmento de Ingeniería Química, Universidad de Huelva, Facultad de Ciencias Experimentales, Campus del Carmen, 21071 Huelva, Spain ‡ Department of Chemical Engineering, Laboratory of Applied Rheology and Polymer Processing, Katholieke Universiteit Leuven, Willem de Croylan 46, B − 3001 Heverlee, Belgium ABSTRACT: Drilling fluids are multicomponent emulsions and/or suspensions, which normally show non-Newtonian behavior, i.e. yield stress, shear-thinning and thixotropy, and strong thermal and pressure dependence. The rheological characterization of drilling fluids using conventional geometries can be a difficult task due to the inherently heterogeneous nature of these systems. These problems may be overcome using nonconventional geometries, such as helical ribbons and blade turbines, which maintain the homogeneity of the system during the measurement. The overall objective of this work was to evaluate the use of mixing geometries, such as helical ribbons and blade turbines, for characterizing the flow behavior of drilling fluids as a function of pressure. From the experimental results it may be concluded that, using the Metzner-Otto approach, both four-blade and helical ribbon type-geometries are suitable for this purpose. This study shows that the Metzner-Otto constant, for a four blade turbine geometry, is practically independent of the flow index at atmospheric pressure, which shows a linear dependence at higher pressures. On the contrary, for the helical ribbon geometry, an exponential dependence of the MetznerOtto constant on the flow index is observed independently of the measured pressure. From the experimental results obtained, it can be concluded that both nonconventional geometries can be used to measure the influence of pressure on the rheological parameters of non-Newtonian fluids. These tools extend the experimental shear-rate window covered by the coaxial cylinders conventional geometry to lower values, allowing the measurement of important engineering parameters, such as, for instance, yield stress.

■

INTRODUCTION Drilling fluids are used in oil well drilling operations to extract the oil from the reservoir. Due to their nature (emulsion/ suspensions of various constituents), these fluids are usually classified as “complex”, that is they show non-Newtonian behavior, such as shear-thinning and thixotropy, and strong thermal and pressure dependence. Substantial attention has been paid over the years to the design and control of drilling fluid rheology, as a result of the impact of rheology on numerous aspects of the drilling and completion process.1−3 The behavior of the drilling fluids under high pressure and high temperature (HPHT) is extremely important for drilling deep wells where these extreme conditions are typical. The characterization of drilling fluids, especially at high pressure and temperature, is not a straightforward task. In general, oil-based drilling fluids, such as the one formulated in this work, display a complex rheological behavior at elevated pressures and temperatures. Moreover, the rheological characterization of this type of fluid, that frequently shows a yield stress in the low shear rate region, can be a difficult task in extreme conditions but of major interest to the oil drilling industry in the design of circulating and transport operations of the suspension inside the borehole.2 Due to the inherent heterogeneous nature of the system (dispersion of solid particles of different sizes), the characterization of drilling fluids based on conventional rheometry can be problematic. Sometimes, erroneous results may be obtained as a consequence of phenomena such as phase separation by settling, wall slippage, destruction of organized media, and, © 2012 American Chemical Society

occasionally, fouling of the measuring gap. As has been reported, when using, for instance, Couette or cone and plate geometries, the ratio of the particle size to the gap width must be smaller than 0.1 to avoid particle-wall interactions.4 This criterion is difficult to meet in drilling fluids with coarse grain suspensions. Solid-wall interactions are hardly avoidable within the small diameters typical of capillary rheometers. Such interactions affect the estimation of the medium material resulting in uncertainty in the flow boundary conditions near the walls.5 Several methods have been proposed in the literature to tackle the rheological characterization of nonhomogeneous fluids. Klein et al.6 proposed a modification of the double cylinder geometry, based on the settling time of particles. Although this new arrangement does not require prehomogenization of the sample, the effect of the settled particles on the torque measurements is difficult to correct. Other nonconventional rheometric alternatives include the use of mixing devices of various kinds fitted to a rheometer instead of the classical viscometric tools. The idea is to apply the basic principles used in the derivation of mixer power curves to estimate the viscosity in the vessel bulk.7 Nonconventional geometries, such as helical ribbons and blade turbines, may be valuable tools for characterizing the rheological behavior of dispersed systems,8 such as drilling Received: Revised: Accepted: Published: 14399

July 11, 2012 October 19, 2012 October 19, 2012 October 19, 2012 dx.doi.org/10.1021/ie301835y | Ind. Eng. Chem. Res. 2012, 51, 14399−14407

Industrial & Engineering Chemistry Research

Article

Two nonconventional geometries, a four flat-blade turbine of 32 mm diameter (FL-4) and 40 mm blade length (L) and a double helical ribbon of 36 mm diameter (DHR-2) and 78 mm impeller length, were supported and centered between two sapphire bearings at the top and bottom of the pressure cell. The clearance between the vessel bottom (h) and the impeller blades was 14 mm in both cases. The most important geometrical parameters are described in Figure 1. The geometrical details of the mixers used in this work are reported in Table 1. The torque acting on the geometries was transmitted from the drive of a controlled-stress rheometer MARS II (Thermo, Germany) by a magnetic coupling. All samples in the pressure cell were tested at 20 °C (±0.1 °C). The temperature in the vessel was maintained constant by circulating a heating fluid from a thermostatic bath. The pressure cell was pressurized by a hydraulic system, using the fluid to be tested as pressurizing liquid. This pressurization system consisted of two units, a high pressure valve and a hand pump (Enerpac, USA), connected by a high pressure line. Samples were carefully introduced from the valve until all the air bubbles inside the vessel were completely expelled. A pressure transducer GMH3110 (Gresingeg Electronic, Germany), capable of measuring pressure differences up to 400 bar (0.1 bar resolution), was used. For calibration purposes, a commercial glycerin (Guinama, Spain) was used as Newtonian fluid. In addition, power-law fluids with different flow indexes, from 0.1 to 0.6, were tested (denoted as A, B, C, and D). In this sense, different aqueous solutions of carboxymethyl cellulose (CMC) (Sigma-Aldrich, Germany), guar gum (Guinama, Spain), and xanthan gum (Guinama, Spain) were prepared. The aqueous polymer solutions were prepared by mechanical agitation, at room temperature, until complete homogenization of the samples. The power-law parameters of these shear-thinning fluids are shown in Table 2.

fluids, mainly in the cases of flow instability phenomena.9 These special geometries can be calibrated using well-known procedures, such as the Metzner-Otto method,10 which has been extensively used for the design of mixers at atmospheric pressure.4 This method, based on a Couette flow analogy,11,12 originally postulates a linear relationship between the so-called average (effective) shear rate around the impeller and the impeller rotational speed, through the definition of the Metzner-Otto constant, Ks, which is a function of the geometry of the impeller. Nevertheless, some authors suggest that Ks is also dependent on flow properties, i.e. power-law flow index, n.13,14 The overall objective of this research was to formulate and characterize drilling fluids with suitable rheological properties. It focuses on the use of nonconventional geometries, such as helical ribbons and blade turbines, for characterizing the flow behavior of drilling fluids as a function of pressure.

2. MATERIALS AND METHODS Mixing rheology measurements were performed in a pressure cell D400/200 (diameter D = 39 mm and height H = 140 mm). This cell consisted of a nonbaffled static vessel with a cylindrical closed head and truncated conical bottom (Figure 1). The cell was completely filled with the fluid to be tested.

Table 2. Power-Law Parameters for the Testing Fluids Used, at 20 °C and Atmospheric Pressure rheological properties fluids

k (Pa·sn)

n

γ̇ (s−1)

glycerin fluid A fluid B fluid C fluid D drilling fluid

1.20 22.67 11.74 34.66 9.87 6.86

1 0.11 0.25 0.45 0.54 0.10

5−100 0.5−100 0.3−10 5−70 2−50 0.1−2.5

Figure 1. Geometric characteristics of the mixing geometries used.

An oil-based drilling fluid (ρ = 930.9 kg/m3 at 20 °C) for testing was formulated by dispersing 5% w/w organoclay, Bentone128 (Elementis, Belgium), in naphthenic oil.

In addition, two conventional coaxial cylinder geometries, PZ (d = 38 mm) and Z (d = 41 mm), from Thermo (Germany) were used to validate the flow properties of the fluids used.

Table 1. Main Geometrical Characteristics of the Mixing Tools Used geometrical characteristics mixing tool

d (m)

L (m)

d/D

w/d

c/d

p/d

L/H

N (rps)

DHR-2 FL-4

0.036 0.032

0.078 0.040

0.920 0.820

0.167 -

0.042 0.109

2.167 -

0.560 0.290

0.008−5 0.008−5

14400

dx.doi.org/10.1021/ie301835y | Ind. Eng. Chem. Res. 2012, 51, 14399−14407


Article

Figure 2. Power number versus Reynolds number for the model fluids studied, at different differential pressures.

relationship (eq 2) into eq 1 and redefining the Reynolds number as

The organoclay suspension was prepared using a high shear rotor-stator mixer Ultraturrax (Ika, Germany) at 9000 rpm, for five minutes, at 120 °C. The rheological characterization of the fluids was performed in the above-mentioned rheometer, MARSII (Haake, Germany), using both conventional and mixing geometries.

Reg =

(1)

where Kp is the power constant. In the case of Newtonian fluids, power consumption depends on both fluid viscosity and impeller geometry. For the impeller geometries studied, the values of the power constant (Kp) obtained were 1465 and 1413 for DHR-2 and FL-4, respectively. 3.1.2. Non-Newtonian Fluids. In the case of non-Newtonian fluids at atmospheric pressure, the influence of shear-thinning on power consumption may be evaluated by using the MetznerOtto approach γav̇ = KsN

deq =

(2)

where Ks is the Metzner-Otto constant, γ̇av is the average shear rate, and N is the rotational speed of the impeller. The average shear rate value is related to the impeller diameter (d), the measured torque (MnN) and the viscous flow properties of the fluid, consistency index (k), and flow index (n), as follows: ⎡ 2πM ⎤1/(n − 1) nN ⎥ γav̇ = ⎢ ⎣ kKpNd3 ⎦

D ⎡ ⎢1 + ⎣

4π N n

πkLD2 2MnN

1/ n ⎤n /2

( )

⎥ ⎦

(6)

As can be seen in the above equation, the equivalent diameter is estimated on the basis of the external diameter of the tank, D. Equation 6 confirms that the equivalent diameter, deq, depends on the rheological properties (k, n) of the fluid. Carreau et al.14 found that the virtual cylinder diameter, deq (n), varies slightly with n for helical ribbon impellers, which means that the Metzner-Otto constant, Ks, should be dependent on n for this type of geometry. Actually, this method models the real flow pattern of a complex geometry in the well-characterized shear rate profile of two coaxial cylinders. In order to confirm this approach and verify how the equivalent internal diameter, deq, is sensitive to the nature of the fluid, torque-angular velocity values for both Newtonian and non-Newtonian fluids were also

(3)

For a power-law fluid, the average viscosity is estimated from ηav = kγav̇ n − 1

(5)

Figure 2 shows the master power number curves, for the two nonconventional geometries analyzed, for all the non-Newtonian fluids (with flow indexes below 0.6) studied, at atmospheric pressure. The Metzner-Otto approach assumes that the fluid motion in the vicinity of the impeller can be characterized by an average shear rate, which is linearly related to the rotational speed of the impeller. Originally, the Ks constant was postulated as a true constant for a given geometry and independent of fluid properties.15 Nevertheless, the Couette analogy is an analytical method which has been proposed to predict the dependence of Ks on geometrical of the impeller and the rheological properties. To find out this relationship, the Couette analogy consists of determining the internal diameter, deq, for an equivalent virtual cylinder, which has the same height L as the impeller, assuming the same torque at the same rotational speed.16 Assuming a steady-state laminar regime and isothermal conditions in the virtual Couette geometry, the equivalent diameter for a power-law fluid is given by

3. RESULTS AND DISCUSSION 3.1. Mixing Rheology at Atmospheric Pressure. 3.1.1. Newtonian Fluids. Power consumption at atmospheric pressure is calculated by the dimensionless power number, Np, as a function of the Reynolds number, Re. In the laminar regime (Re < 10), this relationship is expressed as follows Np = K pRe−1

ρN 2 − nd 2 kKs n − 1

(4)

Thus, a generalized power curve can be obtained for any nonconventional geometry by introducing the Metzner-Otto 14401



Article

0.70. Additionally, the r* value for FL-4 (1.87 × 10−3 m) is only slightly lower than the value expected (1.88 × 10−3 m). Finally, r* can be substituted into eq 7, yielding the following expression for KS:

measured in a coaxial cylinders geometry (PZ). Table 3 illustrates the equivalent diameters for the above-mentioned Table 3. Equivalent Couette Diameters for Conventional and Mixing Geometries As a Function of Fluid Flow Index

⎡ ⎤ deq 2 ⎢ ⎥ 2r * ⎢ ⎥ Ks = 4π ⎢ ⎛ deq 2 ⎞ ⎥ ⎢ ⎜1 − ⎟⎥ D ⎠⎦ ⎣⎝

( ) ( )

equivalent Couette diameter, deq (m) n

PZ

1 0.55 0.45 0.25 0.11 0.10

3.80 3.83 3.85 3.87 3.89 3.87

× × × × × ×

DHR-2 10−2 10−2 10−2 10−2 10−2 10−2

3.53 3.54 3.59 3.73 3.85 3.83

× × × × × ×

10−2 10−2 10−2 10−2 10−2 10−2

FL-4 3.62 3.72 3.74 3.84 3.89 3.84

× × × × × ×

10−2 10−2 10−2 10−2 10−2 10−2

3.2. Mixing Rheometry at High Pressure. 3.2.1. Effect of Pressure on Power Consumption for Newtonian Fluids. As is well-known, most fluids show an increase in viscosity with pressure at constant temperature. For low and medium pressures (up to 1000 bar), the effect of pressure on viscosity can be satisfactorily described by an exponential equation, such as the Barus model19

coaxial cylinders tool, as well as for the nonconventional geometries used, in a wide range of flow indexes. As can be observed, the virtual cylinders obtained for both nonconventional geometries, DHR-2 and FL-4, show a weak shearthinning dependency. Similar results have also been obtained by Ait Kadi et al.17 for a conventional geometry (double Couette). It should be noted that, for all geometries, deq increases as the flow index decreases. Moreover, for highly shear-thinning fluids, the equivalent diameter for both geometries has a limiting value very close to the external diameter of the tool. Taking these results into account, it can be assumed that deq variations are sufficiently small, and the Couette analogy can therefore still be used. Thus, deq(n) = deq(n = 1) is a valid approximation for each geometry. Once deq has been determined from the power consumption data of the Newtonian fluid, the shear rate profile, as a function of position in the gap, r, the equivalent diameter deq, and external diameter D, can be evaluated from the following expression: ⎡ ⎤ deq 2/ n ⎢ ⎥ 2r 4π ⎢ ⎥N γ(̇ r , n) = deq 2/ n ⎥ n ⎢ ⎢⎣ 1 − D ⎥⎦

( ) ( )

μ(P) = μ0 ·exp(β(P − P0))

(10)

where μ is the viscosity at the measured pressure, μ0 is the viscosity at the reference pressure, β is the piezoviscous coefficient, P is the pressure, and P0 is the reference pressure (1 bar). In the case of glycerin, the empirical Barus equation describes the viscosity-pressure relationship fairly well from 1 to 390 bar at 20 °C, the reference viscosity being 1.20 Pa·s, and the piezoviscous coefficient being 5.79 × 10−4 bar−1. Figure 2 shows the power curves vs the Generalized Reynolds number, at different pressures, for the mixing geometries used in this study. As expected, a power-law relationship between the power number and the Reynolds number is found for each impeller. As can be seen from Figure 2, high pressure power number curves match the atmospheric pressure power curve. Consequently, the power constant, Kp, is independent of pressure in the range between 1 and 390 bar. These results show that power consumption at high pressure is only a function of the pressure-viscosity relationship of the fluids. Furthermore, this suggests that the efficiency of the impellers studied is not influenced by viscosity variations due to pressure increase. This finding has special interest, from an engineering point of view, for the design and optimization of the mixing operations at high pressure, such as the crystallization process.20 It should be emphasized that the above relationship does not need, a priori, the power constant value of the special geometry in order to validate it at high pressure. 3.2.2. Influence of Pressure on Non-Newtonian Fluids Mixing. Since there is no evidence, in the case of power-law fluids, about the influence of pressure on Kp for both geometries, this effect has been quantified from its influence on the consistency and flow indexes of the fluid. In many cases,

(7)

This equation predicts a maximum shear rate for deq, which depends on n. As pointed out by Ait Kadi et al.,17 there is an optimal position, r*, in the gap for which the deformation rate is nearly independent of n. Hence, the optimal radius can be obtained from the following condition: γ(̇ r *, n) = γ(̇ r *, 1) = γ(̇ r *)

(9)

(8)

The resulting expression confirms that r* is a very weak function of the flow behavior of the fluid for both mixing geometries. For DHR-2, the r* value observed (1.83 × 10−3 m) is very close to ((D/2) + (deq/2))/2, which agrees with the theoretical result obtained by Bousmina et al.18 for deq/D >

Table 4. Power-Law Parameters for the Calibration Fluids Used, As a Function of Pressure power-law parameters ΔP (bar)

100

200

300

390

fluids

k (Pa·sn)

n

k (Pa·sn)

n

k (Pa·sn)

n

k (Pa·sn)

n

A B C D

21.70 11.70 32.30 9.25

0.13 0.26 0.47 0.57

21.60 11.67 32.07 9.39

0.13 0.26 0.47 0.57

21.52 11.76 31.81 9.53

0.13 0.26 0.47 0.57

21.41 12.07 32.49 9.63

0.13 0.26 0.47 0.57

14402



Article

(helical ribbon DHR-2 and turbine FL-4). The Ks values for both geometries, at high pressure, were determined for each shear-thinning fluid, using eq 11. As can be seen, for the helical ribbon (DHR-2), Ks exponentially increases with the flow index, as expected.24 For the turbine (FL-4), a linear increase in Ks with the flow index is noticed at high pressures, while the value is almost constant (±5.25%) at atmospheric pressure (see Figure 3). It is worth mentioning that Thakur et al. work,11 using the Metzner-Otto method, establishes a correlation of this constant with the d/D ratio, ranging from 0.63 to 0.94. This expression predicts a similar value for Ks (41.7), although a bit lower than those obtained for the four flat-blade turbines (46.8) at atmospheric pressure. In the case of double helical ribbon (DHR-2), the Ks values obtained are slightly higher than those obtained for FL-4 for large flow index values (above 0.4). For this double helical ribbon geometry, Ks dependence on the flow index is described, between 1 to 390 bar, by the following equation fairly well

the consistency index increases exponentially with pressure, and, consequently, power consumption increases, as in the case of Newtonian fluids. On the other hand, pressure may influence the shear-thinning flow index as well. As is well-known, the value of the Meztner-Otto constant may depend on the flow index of the fluid, and, consequently, this dependency may be affected by changes in pressure. Hence, by combining eqs 2 and 3, the Metzner-Otto constant as a function of pressure can be calculated introducing the pressure dependence on both torque consumption and the power-law viscous parameters of the fluid (k, n) to yield the following expression ⎡ ⎤1/(n(P) − 1) π 2 M ( P ) nN ⎥ Ks = ⎢ ⎢⎣ K k(P)N n(P)D3 ⎥⎦ p

(11)

where MnN (P) is the torque applied for each nonconventional geometry, and k(P) and n(P) are the consistency and flow indexes of the non-Newtonian fluid at the measured pressure (see Table 4). The power-law fluids prepared for calibrating the nonconventional geometries studied were characterized by using coaxial cylinder geometries, in the pressure range of 1−400 bar, at 20 °C. The results are presented in Table 4. As can be seen, the effect of pressure on the power-law parameters of these fluids is very weak. A similar behavior has been encountered in aqueous dispersions and/or solutions, at the same temperature, in the pressure range from 1 to 400 bar,21 where the effect of pressure on the rheological behavior is defined by the solvent characteristics.22,23 Figure 3 shows the evolution of the Metzner-Otto constant, Ks, with the flow index, for both mixing geometries studied

Ks = Ks0 + b·exp(αn)

(12)

where Ks0 is the minimum value of the Metzner-Otto constant (40.5) in the limit of very low flow index value, and b and α are equal to 1.70 and 4.35, respectively. Thus, this expression predicts a finite Ks value for highly shear thinning fluids (n → 0), with a positive increase in Ks as the flow index increases. Other authors, such as Brito et al.,24 predict a value of zero. In addition, Delaplace et al.15 have developed a method quite similar to the Couette analogy. The prediction of Ks for analogous geometries, such as DHR-2, using this model show values lower than those obtained in this work, in range of flow index values tested. The difference in the Ks dependence on shear-thinning effects is most likely to be accounted for by errors in the determination of Kp, as is pointed out in Delaplace et al. work. On the other hand, the flat-blade turbine geometry, FL-4, shows a slight pressure dependence of Ks. Thus, as can be seen in Figure 3, average shear rates (or Ks values) are increasingly sensitive to pressure as the flow index increases. For this impeller, a more complex expression was needed to correlate Ks experimental data with pressure. It is important to note that the particular influence of pressure on FL-4 mixing behavior may be due to the pattern flow effects outside the gap, as will be discussed later on. The following equations provide a close fit with the combined effects of both pressure and index flow on Ks for FL-4

Figure 3. Evolution of Ks with the flow index for the mixing geometries used. Ks (n) are estimated from eq 12 for DHR and from eqs 13 and 14 for FL, respectively.

Ks = Ks0 + d(P) ·n

(13)

d(P) = A + B(P − P0)

(14)

where Ks0 is a constant in the pressure range studied, (43.8 ± 3%), and d(P) is a parametric function of pressure with two

Table 5. Equivalent Diameters for Conventional and Mixing Geometries at High Pressure equivalent Couette diameter, deq (m) ΔP (bar)

200

390

n

0.13

0.47

1

0.13

0.47

1

PZ DHR-2 FL-4

3.85 × 10−2 3.84 × 10−2 3.89 × 10−2

3.85 × 10−2 3.58 × 10−2 3.73 × 10−2

3.80 × 10−2 3.53 × 10−2 3.62 × 10−2

3.86 × 10−2 3.84 × 10−2 3.89 × 10−2

3.84 × 10−2 3.58 × 10−2 3.72 × 10−2

3.80 × 10−2 3.53 × 10−2 3.62 × 10−2

14403



Article

fitting parameters, A and B, having values of 4.95 and 0.04, respectively. On the basis of the Couette analogy, Ks values can be estimated by substituting the r* value for each geometry in eq 9. Thus, the analysis of the equivalent diameter, deq, is essential in order to elucidate how pressure might influence shear rate in the gap. Table 5 reports the equivalent diameter, calculated according to the Couette procedure, for the geometries used at high pressures. As can be observed, pressure does not have significant effects on the equivalent diameter for the geometries tested. Consequently, these results indicate that the optimal radius r* is kept constant for all non-Newtonian calibrating fluids between 1 to 390 bar. Furthermore, both mixing geometries yield a shear pattern in the gap that apparently does not change when the rheological response of the fluid is not largely dependent on pressure. According to this procedure, the Ks value calculated for the flat-blade turbine geometry, FL-4, is considerably overestimated (79.1) compared to the experimental values, which range between 43.8 and 54.2, from 1 to 390 bar. By contrast, the predicted value for DHR-2 (58.7) is in relative agreement with the experimental Ks values obtained (40.5−62.3) in the flow index range studied. These results seem to indicate that, for the double helical ribbon geometry, the flow patterns would be similar to those generated by a conventional coaxial cylinder but not in the case of a fourblade geometry.5 Similarly, it is worth pointing out that the Couette analogy could explain the dependence of the MetznerOtto constant on the flow index for the complex nonconventional DHR-2 geometry. Under the condition of a narrow gap, such as DHR-2, the shear rate, or Ks, can vary with n, showing opposite trend depending on the location at which this parameter is evaluated. In other words, the estimation of the shear rate generated by the impeller could show a strong dependence of n as function of radial position in the gap, as pointed out by Bousmina et al.18 As can be seen in Table 3, the equivalent diameter tends to D as shear-thinning characteristics decrease. Consequently, the flow index influence should be taken into account close to the wall with an acceptable approximation. Thus, in the proximity of the wall, the expected changes in the Ks parameter coincide with the experimental variation observed when the Metzner-Otto method is used.18 However, FL-4 shows the same equivalent diameter trend, but the Metzner-Otto method and the coaxial cylinder analogy do not match. The small increase in Ks values obtained for FL-4 could indicate that torque measurements are slightly influenced by the end effect contributions on the axis direction.25 For the flat-blade turbine studied, the small L/H ratio may affect the shear rate generated in the vessel as pressure increases, particularly for the least shear thinning fluid (fluid D). As has been reported elsewhere, the flow pattern of these geometries is totally different,26,27 despite showing similar values for the Metzner-Otto constant. In the case of the double helical ribbon geometry, the fluid between the blades and the wall flows upward, inward along the surface, downward at the core near the shaft, and radially outward near the base of the tank.28 The direction of this primary flow can be inverted by reversing the impeller rotation. By contrast, the main flow pattern in a flat-blade agitator basically consists of an imaginary cylinder where the high shear rates are concentrated in a narrow annulus circumscribed close to the tips of the blades.29 As can be seen in Figure 3, both mixing geometries show comparable Ks values in the range of the pressure investigated for the shear-thinning window scoped. Thus, these results

confirm that both geometries are strongly dependent on the clearance of the mixer system, as expected.30−32 This seems to indicate that the ratio d/D, independent of the impeller geometry, is the controlling factor in generating high shear values in the annulus gap for close-clearance geometries like FL-4 and DHR-2 (see Table 1). 3.2.3. Rheological Measurements of Non-Newtonian Fluids at High Pressure. Figure 4 shows the steady-state flow

Figure 4. Viscous flow curves, for selected calibration fluids, at atmospheric and high pressure. Comparison between experimental data obtained from coaxial cylinders, double helical ribbon, and flat blade turbine geometries.

curves for fluid B (n ∈ 0.25−0.26) and fluid D (n ∈ 0.55−0.57) using conventional and mixing geometries, at 20 °C, at two different pressures (1 and 390 bar). As can be observed, the resulting viscous flow curves for both fluids agree match the results obtained under viscosimetric flow conditions with the coaxial cylinder used (PZ). A similar behavior was observed for all the calibration fluids tested, at pressures between 1 and 390 bar. It is important to note here that the average viscosity and average shear rates from DHR-2 and FL-4 were obtained after processing both torque and rotational speed data by means of eqs 3 and 4. The excellent agreement between flow curves from both conventional and mixing geometries is only noticed when a flow index-dependent relationship, Ks(n), is used for these mixing geometries. Additionally, it should also be noted that the estimated viscosity values obtained using a constant Ks value from the Couette analogy for DHR-2 has higher positive deviation for a strong shear-thinning fluid (fluid B) than for a weak shear-thinning fluid (fluid D), as expected from the Ks values obtained. A similar behavior has also been reported by some authors5,33,34 when comparing data from conventional and nonconventional geometries, such as helical ribbon and anchor impellers, coupled to a rheometer. Finally, these rheomixers have been used for characterizing a typical drilling fluid at high pressure. Power-law parameters of the drilling fluid tested are presented in Table 6. As can be seen, both power-law parameters show a clear pressure-dependence from 1 to 390 bar. This behavior is frequently encountered in drilling fluids having a viscosity-pressure relationship essentially dominated by the continuous media.35 Figures 5 and 6 show a comparison of the steady-state viscous flow curves of the drilling fluid, obtained by using a coaxial cylinder tool, a double helical ribbon geometry (Figure 5), and a flat-blade turbine geometry (Figure 6), at 1, 200, and 390 bar. As can be observed in both figures, the flow curves 14404



Article

of torque vs rotational speed, using both mixing geometries, for the drilling fluid studied.

Table 6. Drilling Fluid Power-Law Parameters at High Pressure power-law parameters ΔP (bar)

k (Pa·sn)

n

100 200 300 390

9.33 10.05 11.05 12.09

0.14 0.18 0.20 0.24

Figure 7. Evolution of torque with impeller rotational speed for the drilling fluid studied, by using coaxial cylinders and mixing geometries, at different pressures.

As can be observed in this figure, the drilling fluid tested shows a tendency to a limiting value of the torque at low rotational speeds, which corresponds to an apparent yield-stress at low shear rate. Similarly, it can also be seen that the flat-blade turbine geometry yields higher fluctuations in torque than DHR-2, between 0.01 and 0.1 s−1. This result suggests that the shear distribution in the DHR-2 geometry is more homogeneous than in FL-4, as a consequence of a higher L/H ratio, the former being more sensitive to capturing the rheological properties of these fluids in the low shear region, such as the above-mentioned apparent yield stress. Figure 8 shows the apparent yield stress, τc, estimated from the fitting of the Bingham model to the experimental data

Figure 5. Viscous flow curves for the drilling fluid studied, at atmospheric and high pressures, obtained by using coaxial cylinders and double helical ribbon geometries.

Figure 6. Viscous flow curves for the drilling fluid studied, at atmospheric and high pressures, obtained by using coaxial cylinders and flat-blade turbine geometries. Figure 8. Influence of pressure on drilling fluid yield stress, at 20 °C.

obtained from the conventional and the two mixing geometries agree fairly well in a shear-rate range from 0.1 to 2.0 s−1. Thus, both mixing geometries match the pressure dependence of viscosity, obtained with conventional geometries, for this drilling fluid, even though the shear-thinning behavior varies over the whole pressure range studied (1−390 bar). Additionally, it must be emphasized that these mixing geometries extend the experimental range of shear rate down to a much lower shear-rate region, showing that pressure has a little influence at the lowest limit of measurable shear rates. In addition, this result proves that mixing geometries can be satisfactorily used in order to deal with flow instabilities that drilling fluids show below a critical shear rate,1 even in high pressure environments. This fact is evidenced in Figure 7, which depicts the evolution

presented in Figure 8, obtained with the double helical ribbon geometry. As can be observed, yield stress evolves linearly with pressure (between 1 to 390 bar)

τc = τc0 + βτ (P − P0)

(15)

where τc0 is the value of yield stress at atmospheric pressure, and βτ is the piezo-yield-stress coefficient. The values for τc0 and βτ are 3.75 and 0.041 Pa·bar−1, respectively. These results indicate that pressure has a small effect on the rheological properties of drilling fluids, increasing both yield stress and viscosity. The complex viscosity-pressure dependence of these suspensions may be attributed mainly to physical changes 14405



■

induced by compression of the continuous oil phase under high hydrostatic pressure and, consequently, to an increase in the effective fraction volume of the disperse phase.2,36

4. CONCLUSIONS In this study, the influence of both shear-thinning behavior and high pressure on the power consumption of two impellers for complex fluids mixing was investigated. Two methods, the classical Metzner-Otto procedure and the Couette analogy, were used to describe the relationship between power input data and shear-rate. From the Newtonian data obtained at high pressure, it is apparent that the power consumption only depends on the viscosity-pressure relationship of the fluid. Additionally, this study shows that the Metzner-Otto constant, for a four blade turbine geometry, is practically independent of the flow index of the fluid at atmospheric pressure, while it shows a linear dependence at higher pressures. On the contrary, for the DHR-2 impeller, an exponential dependence of Ks on the flow index is observed independently of the measured pressure. The proposed expressions for calculating the Ks values of both nonconventional geometries are useful to predict the influence of shear-thinning behavior and pressure. The clearance seems to be the key variable to determinate the Ks values, despite being completely different mixing geometries. Based on the Couette flow analogy, flow patterns of both geometries are not influenced by pressure changes. In addition, under high pressure, Ks values predicted by the Couette analogy are in agreement with that obtained using the Metzner-Otto procedure for DHR-2, whereas values are overestimated for FL4. As a result, the Couette approach is a valid approximation to explain the variation of the experimental Ks values as function of pressure and flow index for the double helical ribbon geometry. The viscous flow curves of single phase non-Newtonian fluids for both nonconventional geometries, applying the Metzner-Otto method, practically coincide with those obtained using the coaxial cylinder geometry in the range of pressure studied. Moreover, using the value of Ks predicted by the Couette analogy for DHR-2 allows a good estimation of the viscosity-shear curves of non-Newtonian fluids in a wide range of pressure. The two mixing geometries studied are suitable for characterizing the flow behavior of dispersed systems, such as drilling fluids. These tools extend the experimental shear-rate window covered by the coaxial cylinders conventional geometry to lower values, allowing the measurement of important engineering parameters, such as, for instance, yield stress. This type of mixing geometries, mainly DHR-2, offers a very useful tool to relate microstructure and rheology of these suspensions in a high pressure environment.

■

Article

NOTATIONS A = fitting parameter (eq 14) b = pre-exponential parameter (eq 12) B = fitting parameter (eq 14) c = wall-clearance ((D − d)/2) (m) d = impeller diameter (mm) deq = equivalent diameter (m) d(P) = parametric function of pressure (eq 13) D = vessel diameter (m) h = bottom clearance (m) H = height of the vessel (m) k = consistency index (eq 4) (Pa·sn) Ks = Metzner-Otto constant (eq 2) Kp = power constant (eq 1) L = length of the impeller (m) MN = Newtonian torque input (N·m) MnN = non-Newtonian torque input (N·m) n = flow behavior index (eq 4) N = rotational speed of the impeller (rps) Np = power number p = helix pitch P = measured pressure (bar) P0 = reference pressure (1 bar) r = radial coordinate (m) r* = optimal radial position (m) Re = Reynolds number (eq 1) Reg = Generalized Reynolds number (eq 5) w = ribbon width (m)

Greek letters

α βτ γ̇av η ηav μ μ0 ρ τc τc0

■

exponential fitting parameter piezo-yield-stress coefficient (eq 15) (Pa·bar−1) average shear rate (eq 2) (s−1) non-Newtonian viscosity (Pa·s) average non-Newtonian viscosity (Pa·s) Newtonian viscosity (Pa·s) Newtonian viscosity at pressure and temperature of reference (Pa·s) density (kg·m−3) yield-stress (eq 15) (Pa) yield-stress at atmospheric pressure (eq 15) (Pa)

REFERENCES

(1) Coussot, P.; Bertrand, F.; Herzhaft, B. Rheological Behaviour of Drilling Muds, Characterization Using MRI Visualization. Oil Gas Sci. Technol. 2004, 59, 23. (2) Darley, H. C. H.; Gray, G. R. Composition and Properties of Drilling Fluids and Completion Fluids, 5th ed.; Gulf Professional Publishing: Houston, TX, 1988. (3) Luckham, P. F.; Rossi, S. The Colloidal and Rheological Properties of Bentonite Suspensions. Adv. Colloid Interface Sci. 1999, 82, 43. (4) Barnes, H.; Edwards, M. F.; Woodcock, L. V. Application of Computer Simulations to Dense Suspension Rheology. Chem. Eng. Sci. 1987, 42, 591. (5) Brito-de la Fuente, E.; Nava, J. A.; López, L. M.; Medina, L.; Ascanio, G.; Tanguy, P. A. Process Viscosimetry of Complex Fluids and Suspensions with Helical Ribbon Agitators. Can. J. Chem. Eng. 1998, 76, 689. (6) Klein, B.; Laskowski, J. S.; Partridge, S. J. A New Viscometer for Rheological Measurements on Settling Suspensions. J. Rheol. (Melville, NY, U. S.) 1995, 39, 827. (7) Guillemin, J. P.; Menard, Y.; Brunet, L.; Bonnefoy, O.; Thomas, G. Development of a New Mixing Rheometer for studying Rheological

AUTHOR INFORMATION

Corresponding Author

*Phone: +34 959219993. Fax: 34 959219983. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS This work has been sponsored by FEDER-Excellence Projects Programme (Research project P08-TEP-3895). The authors gratefully acknowledge the financial support. 14406



Article

Behaviour of Concentrated energetic Suspensions. J. Non-Newtonian Fluid Mech. 2008, 151, 136. (8) Kelessidis, V. C.; Maglione, R. Yield Stress of Water-Bentonite Dispersions. Colloids Surf., A 2008, 318, 217. (9) Cullen, P. J.; Ó Donell, C. P.; Houška, M. Rotational Rheometry Using Complex Geometries-A Review. J. Texture Stud. 2003, 34, 1. (10) Metzner, A. B.; Otto, R. E. Agitation of Non-Newtonian Fluids. AIChE J. 1957, 3, 3. (11) Rajeev, K, T.; Viall, Ch.; Djelveh, G.; Labbafi, M. Mixing of Complex Fluids with Flat-Bladed Impellers: Effect of Impeller Geometry and Highly Shear-thinning Behaviour. Chem. Eng. Process. 2004, 43, 1211. (12) Cordobes, F.; Brito-de la Fuente, E.; Gallegos, C. Effect of Rheological Properties on Power Input and Effective Shear Rates When Mixing with Different Impellers. An. Quim. Int. Ed. 1998, 94, 274. (13) Yap, C. Y.; Patterson, W. I.; Carreau, P. J. Mixing with Helical Ribbon Agitators. AIChE J. 1979, 25, 508. (14) Carreau, P. J.; Chhabra, R. P.; Cheng, J. Effect of Rheological Properties on Power Consumption with Helical Ribbon Agitators. AIChE J. 1993, 39, 1421. (15) Delaplace, G.; Romuald, G.; Leuliet, J.-C.; Chhabra, R. P. An Analytical Model for the Prediction of Power Consumption for ShearThinning Fluids with Helical Ribbon and Helical Screw Ribbon Impellers. Chem. Eng. Sci. 2006, 61, 3250. (16) Choplin, L. In situ Rheological Follow-up of Food Processes: Application to Emulsification and Ice Cream Fabrication Processes. 2nd Intern. Sysposium on Food Rheology and Structure. Zurich, March 2000; Paper 63. (17) Aït-Kadi, A.; Marchal, P.; Choplin, L.; Chrissemant, A. S.; Bousmina, M. Quantitative Analysis of Mixer-Type Rheometers using the Couette Analogy. Can. J. Chem. Eng. 2002, 80, 1166. (18) Bousmina, M.; Ait-Kädi, A.; Faisant, J. B. Determination of Shear Rate and Viscosity from Batch Mixer Data: Theorical and Experimental Results. J. Rheol. (Melville, NY, U. S.) 1999, 43, 415. (19) Barus, C. Isothermals, Isopiestics and Isometrics relative to Viscosity. Am. J. Sci. 1893, 45, 87. (20) Bertucco, A.; Vetter, G. High pressure Process Technology: Fundamentals and Applicatons. Industrial Chemistry Library Vol. 9, 1st ed.; Elsevier Science B.V.: The Netherlands, 2001. (21) Briscoe, B.; Luckham, P.; Zhu, S. Rheological Study of Poly(ethylene oxide) in Aqueous Salt solutions at High Temperature and Pressure. Macromolecules 1996, 29, 6208. (22) Cook, R. L.; King, H. E., Jr.; Peiffer, D. G. High-Pressure of Dilute Polymer Solutions in Good Solvents. Macromolecules 1992, 25, 2928. (23) Sawamura, S.; Yoshimura, Y.; Kitamura, K.; Taniguchi, Y. Effects of Pressure, Temperature, and Concentration on the Viscosity of an Aqueous Solution of Sodium Chloride. J. Phys. Chem. 1992, 96, 5526. (24) Brito-de la Fuente, E.; Leuliet, J. C.; Choplin, L.; Tanguy, P. A. Mixing with Helical Ribbon impellers: Effect of Highly Shear Thinning Behaviour and Impeller Geometry. Trans. IChemE 1997, 75, 45. (25) Savarmand, S.; Heniche, M.; Béchard, V.; Bertrand, F.; Carreau, P. J. Analysis of the Vane Rheometer using 3D Finite Element Simulation. J. Rheol. (Melville, NY, U. S.) 2007, 51, 161. (26) Perse, L. S.; Bajcar, T.; Sirok, B.; Zumer, M. LDA Velocity Measurements of High-Viscosity Fluids in Mixing Vessel with Vane Geometry Impeller. Can. J. Chem. Eng. 2007, 85, 817. (27) Zhang, M.-G.; Zhang, L. H.; Jiang, B.; Yin, Y.; Li, X.-G. Calculation of Metzner Constant for Double Helical Ribbon Impeller by Computational Fluid Dynamic Method. Chin. J. Chem. Eng. 2008, 16, 686. (28) Delaplace, G.; Leuliet, J. C.; Relandeau, V. Circulation and Mixing Times for Helical Ribbon Impellers. Review and Experiments. Exp. Fluids 2000, 28, 170. (29) Yan, J.; James, A. E. The Yield Surface of Viscoelastic and Plastic Ffluids in a Vane Viscometer. J. Non-Newtonian Fluid Mech. 1997, 70, 237.

(30) Takahashi, K.; Sasaki, M; Arai, K. Effects of Geometrical Variables of Helical Ribbon Impellers on Mixing of Highly Viscous Newtonian liquids. J. Chem. Eng. Jpn. 1982, 15, 217. (31) Takahashi, K.; Yokota, T.; Konno, H. Power Consumption of Helical Ribbon Agitators in Highly Viscous Pseudoplastic liquids. J. Chem. Eng. Jpn. 1984, 17, 657. (32) Shekhar, S. M.; Jayanti, S. Mixing of Pseudoplastic Fluids Using Helical Riboon Impellers. AIChE J. 2003, 49, 2768. (33) Jomha, A. I.; Edwards, M. F.; Woodcock, L. V. New Method for predicting the Power Requirements for Mixing Shear Thickening Suspensions. Chem. Eng. Sci. 1990, 45, 1389. (34) Allen, D. G.; Robinson, C. W. Measurement of Rheological Properties of Filamentous Fermentation Broths. Chem. Eng. Sci. 1990, 45, 37. (35) Combs, G. D.; Whitmire, L. D. Capillary Viscometer Simulates Bottom Hole Conditions. Oil Gas J. 1960, 108. (36) Briscoe, B. J.; Luckham, P. F.; Ren, S. R. The properties of Drillings Muds at High Pressures and High Temperature. Philos. Trans. R. Soc. London, Ser. A 1994, 348, 179.

14407


High Pressure Mixing Rheology of Drilling Fluids - American Chemical

Recommend Documents